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Full text of "Nuclear Physics by Enrico Fermi: Chapter 8"


A. dotation 

The nuclear reaction A 

+ a 

■3 + p + <4 is symbolized by 

alpha, p proton, d deuteron, 3 

Pa r t i c ] b s are s jmb o L i z ed by J 
gamm a ray , a nd f for fission. 

Q is (+) for an "exothermic" reaction, (-) for " ondo thermic" . 

ThG thres hold is the minimum energy of the bombarding part- 
icle in order far the reaction to occur. Threshold is measured 
in the laboratory . system, and therefore is not necessarily equal 
in magnitude to Q. If Q is positive, the threshold is, in nrin- 
ciple, 0. If Q is negative, and If the bombarded particle A Is 
approximately at rest, then ( see ph. I, rage 5) 

Threshold energy = (-Q) X Mss_qf incipi e nt par ticle 

Reduced mass of svstem 

= (-Q)X^i^- 


VIII. 1 

for the reaction symbolised above, 

B. Gen er al Features of Cross- s ecti ons for Mr clear React ions. 

The following considerations apply to cross- sections for 
nuclear reactions in the absence. of resonances. Resonance phen- 
omena are discussed in section D. 

Consider the transition A + a— *-B + b + Q, v.here the nuc- 
leus "A" and the particle "a" become the nucleus "E" and particle 
"b" Both the initial and final states of the system consist of 
a pair of unbound particles; therefore the transitions is to one 
of a continuous distribution of states. The initial state also 
has a continuous range of possible energies, but the experiment 
itself specifies a particular Initial energy . 

There are sl;r.Ilar situations in atomic physics. For example, 
In emission of a photon by an excited aton, the transition is from 
a sinr-Ie state to one of a continuum of states: 
Conservation of energy selects the final state. 

Another atomic example is the non- radia- 
tive or Auger transition. An excited atom may 
have two: possible modes of decay. In addition 

CCtom, no 

cttom b dnd-a 

to photon emission, the atom may decay by emis- 
sion of an electron. Suppose, for example, the 
excitation corresponds to one missing electron 
in the K shell . The energy made available 
vahen an electron falls into this hole may be greater than the 
ionization energy, in which case an electron may be emitted from 
the atom. Again the final system consists of tv:o unbound particles 
having a continuous range of possible energies , 

Returning to the nuclear reaction A + a — *-B + b, we use 
a general principle of quantum mechanic s to derive some essen- 
tially statistical results on the variation of the cross- section. 

From quantum mechanics, the probability per unit time of 


14 2 

"Golden Rule No 

Nuclear Reactions 
number of transitions per unit time 


W = 


2 «k 

w is given by 

VIII. 2 

where H is the matrix element of the perturbation causing the 
transition, and dn/dE = energy density of final states, counting 
each degenerate state separately. 

lXi' a may be the same for all energetically possible final 
states; more often it defends on the state. (For instance, \H\ 
may depend on the direction of emission.) Then \H\* in the form- 
ula is a suitable average over the possible final states .**** 

dn/dE = oo for a continuum 

\'Hs\ ^ 0, so that the expression 

form x oo . This difficulty is 
box of volume JX 

of states. But in that case 

1M| 2 dn/dE has the indeterminate 

removed by limiting space to a 
small but" finite and dn/dE large 

|M] is then 
but finite. JL drops out of the result. The number of final 
states equal 3 the number of states of the emitted particle. Thii 
is because a change In momentum of one particle compels a change 
in momentum of the other, by conservation of linear and angular 
momentum of the system. 

It was shown in Chanter IV, o. 76 that the number of states 
available to a free particle, "b" , with momentum between p 
p + dp, confined to a box of volume JTj , is 

VIII. 3 


du = 

This must be multi oiled by the multiplicity in the final state** 
caused by spin orientation, which is given by the factor (2I b +l)x 
(2I B +l), where I b Is the spin of the emitted particle and Ig the 
soln of the nucleus. If b is a photon, (2I b +l) Is put equal to 

dE = v% 

dp^. (true relativistlcally) 

VIII .4 

where p^ and w, are the momentum and velocity in the center of 
mass frame of reference of the final (B+b) state. Since B is 
usually massive compared with "b" , p b and v t can usually be meas 
ured in the laboratory frame. Combining these two equations: 

dn = 4-T^'Jl (2%+lX2T B +0 

VIII. 5 

From this and VIII. 2 we get 
ho. transitions per unit time 

LAa|Hl 2 (2V0(2T B + vill.6 

Tffi 4 ^ 

The following equation is essentially 
section C^v^g per A nucleus: 


the croso- 

* Derived in Schiff, Qu antum Mechanics , p. 193. ("Golden Rule No. l" 

is on oare 148 of this text ) . 

** This is discussed in greater detail In section C, this chapter. 

*** This roint is discussed by Bethe and Placzek, Phys.Rev . J51 

450, Appendix, p. 483. Multiplicity is caused by the two possible 

independent polarizations, 

-&HBH*- See page 214 for more complete discussion. 


Nuclear Reactions 

No. transitions/sec _ Y .,.-- v *f 

per "A" nucleus - '** X J "a"rel . to"A" x G A ^B 

VIII. 7 

where A arid. B refer to the (A+a) and (3+b) states respectively, 
and ru. Is the density of particles "a". Take n^ to "be l-jfe, cm" 3 
(one particle in the volume). Then 

J^ Vjrel.to'A" * a-^-b tt^ /K 

VI II. 8 

Since nucleus "A" is often massive compared to "a", V" a "rel .±o"A" 
is often nearly equal. tfl'T a in the center of mass frame. 'In 
any case, these two velocity magnitudes are related by a cons- 
tant factor. Writing ^"'Vrel .to" A" ~ 1j ~a ' 

A-?-B TT-K 

=Wl^>t| 2 A(2V0t2I B ^) 

n^J h 

VIII .9 

In general, )t is unknown. It has the form J drEfy^ U f in{ ha.l 

where U is the interaction energy*. If the wave functions used to 
comoute M are normalized in volume JL , Sh. disappears from the 
expression \SL H\ in VIII. 9. This is seen as follows: let I have 
the form, at large distances, K exp(ikz). Then y"|Yi a o.r = Nlfi 

Setting ifjl = 1, we get N = l/<fcfi? 

If T, ... -, ard T„ -, now mean the un- normalised Diane wave 
xnj_Tri.a_L i ma_L 

functions, the matrix element factor In VIII. 9 becomes 

sih^ f^T Wl ur lmtal 

VIII .10 

(This may be looked upon as taking Jt= lJ Henceforth we use ~H 
for JlH . In order to show the meaning of this expression, we 
write it as 

pfs£| = [J X Volume of nucleus X ] T mi tial T-fW) 

VIII. 11 

Where H|^r in Is a suitable average of the product of the wave func- 
tions over the volume of the nucleus. U, and hence the integrand, 
is zero outside the nucleus. U = average Interaction energy^ 
depth of potential well. For our purposes here the Important 
feature of VIII. 11. is Its dependence on the charge of the parti- 
cipating "particles. If "a", say ; is positively charged, its wave 
function will be reduced in amplitude at the nucleus by the 
barrier factor exp(-G- a /2), where, by III. 3, p. 5$. 


— c barriers 


U & denotes the charge of "a" times the Coulomb potential of "A". 
Physically this factor represents Coulomb repulsion. The wave 
function of an outgoing particle at the- nucleus is also reduced 
by such a barrier factor. The result for the squared matrix ele- 
ment is: it— \ 2 

For neutral particles: IX I <£(u X Vol. of nucleus) VI 1 1. 13 
For + charged oartlcles: |Hi*&(U X Vol .)*X exH-G^-Gj,) 


Nuclear Reactions 

Ob... VIII 

femission of negatively charged particles (elect rons)ls treated 
in Ch, IV ) 

For endo thermic reactions there is a threshold energy for the 
bombarding particle. For exothermic reactions in which the energy 
liberated is much larger than the energy of the bomoarding particle, 
there are tm simplifications in equation VIII .9= l) the barrier 
factor exc(-G n ) for the outgo lrig particle is almost constant be- 
cause it is a function of energy of the emitted particle u , 
which is almost constant; 2) p^ and v b are almost constant and 
therefore the statistical weight factor In VIII. 9, Pt/ V a v b* lB 
proportional to l/T^f 

These results are now apolied to specific cases to deduce the 
general features of the vs. energy and d vs. velocity curves. 

1) ELASTIC (n,n) (both particles , uncharged) 

. ./^ 'therefore" p|/v v, = C*Wt ) > a conBtant 
At low energy \H\ Is approximately constant, therefore 

constant at low energy. 

ELASTIC (n,n) 


a le^Y e.v. 

0f a 

EKOTHEgMl'G. low energy UNCHARGED bombarding particle, as In 
(n,.a), ( n , p } , [n , if, ( n , f ) . 4 is usually ,~ Mev «hl le neut- 
ron energy is— e.v., therefore v b ^ constant . Therefore 
p b Vv a v b Z lAa • r*l E °C exp ( - G n - %1 . exp (- % } is . & c onstant , 
since it den ends on the almost constant energy of the out- 
going particle, or, in the case of an uncharged b , is 1 
exactly. Also exp(-G r ) = 1. Therefore 
a ~ 1/T„ (the "1/v" law) 



3) INELASTIC (n,n' ) 

The nucleus Is left in an excited state. The process is enao- 
thermic and -<J is the excitation energy of the nucleus. 5 or 
Ircident neutron energies slightly above the threshold, v e=j 
constant, since the fractional change In incident energy Is 
small. Put v , changes relatively greatly in this region: 

v t (C excess o 
n <c 

Pn' / T n v n' 

f energy above the threshold. Therefore 
,. v j i oc ^ energy excess 1 . Therefore near the 

threshold cS CC \| energy excess. 

INELASTIC iri,v') cr 


-T We s hj Id 
[ ¥ neu TVoN 

4 ) ENDO TIIERMI C , CHARGED OUT GO I NG part I c 1 a , as In ( n , a ) , ( n , p ) 

3) except t^at the factor exp(-Gv,J 

Exactly as In case _ 

operates and is d o m 1 nant . ^ e&pnersy excess 1 X exp( 


Inverse Processes 


5) EXOTHERMIC, CHARGED INCOMING particle, as in (p,n), (a,n), 
(a.,'3 ), (p, "tf ). For incident energies << Q, the factor 

p^ /v a T-^ oC l/v a . The barrier factor exp(-G- a ) operates on 
the incoming particle. 


<y cC l/v„ s&cpC-Ga) 

In all of the above, no account has been taken of resonance 
phenomena . 

■ Inve rs e Processes 

Consider the transition A + a — >- B + b, where "A" and M B I: 
are nuclei and "a" and "b" are, In general, lighter particles. 
From equation VIII. 9 the cross- section for this transition is 
(neglecting spins): , , 

A _- JffUHr Jt- vt-.,.., 



7T*l 4 


The Inverse reaction Is B + b — > A + a. 


cr Q 



Its cross section is 

VIII. 9" 

|uLr*l is the same In both cases, because the operator of the 
perturbation is Hermit ian, i.e., I ' : fti*X¥&£\ = \ (w'* U V (tfc\ 



Cs^-A ft* 

(neglecting spin) VIII. 14 

The same result may be looked at from a different aspect. 
Suppose we have a box filled with arbitrary numbers of particles 
"A", "a", "B" , "b" . The transitions A + a ^=t B + b occur. 
Statistical mechanics asserts that at equilibrium all possible 
states of the system consistent with the specification of the 
energy of the system are occupied with equal probability. If a 
state consisting of a pair of particles ~A + a is called an "A" 
state, and similarly for "B" state, then the occuoied states in 
the energy range AE may be divided into the two' tyoes , A and B. 
Since all states in AE are equally probably occupied, this div- 
ision Is such that 

No . occupied A states 



possible A states in A E 


occupied B states No. possible B states in AE 
The number of possible A states = maximum number of (A + a) pairs 
times the number of states In A E for one oair - 7? 4"rr fa. _& 



where '( - maximum number of (A + a) pairs formable With the 
particular numbers of particles cut into the box Initially. Sim- 

ilarly, the number of possible B states 
where J} is the same. Therefore 



No. of 

Inverse Processes 

of o ccupied A states __ Pi ^b 
c cue led B states 


VIII .16 

mw at equilibrium the number of transitions A-^B equals the 
number of transitions in reverse, per unit time. 

No. transitions A-^B/sec = (Ko 
No. transitions B^-A/sec = (No 

Combining VIII. 16 with VIII. 17, 

A states occupied ) A ^ g ^ 

B states occupied )CT e ^ fir to 

VIII, 17 

VIII. 14' 



as "before 

If the particles have spins, the density of state 3 is in 


If" the spins are I A , I a , I B , I b , the density of A 
states is increased by the factor (2I A +1 ) (2I a +l ) , and similarly 
for 3 str-t.pp Thfin t/hft rate of transition A— i-B is proportional 

"tes. Then the rate of transition 
(2I A+ l)(2I a+ l)p a 2 C^ B 


-A to 

<2i 3 +i) (2i- 3+ i) Pb 2 cr e ^ A 



, <U B = (2^+0(2^0^^ 


Note that in this formula, O is an average over the various kinds 
(spin orientations) of A states, and a sum of partial Cf s for 
various possible final states. * 

* This may be elucidated by writing O^g more explicitly. Divide 
C Into contributions Ots) due to various relative orientations 
of I. and I n . The number of states represented by each relative 
orientation is 2Sf 1 , where S = resultant angular momentum of part- 
icles "A" and "a". In this discussion, orbital angular momentum 
Is neglected. It is Included in a discussion in the appendix of 
Bethe and Placzek 1 s paper. Phys.Rev . SJL 450. The total number of 
A states Is (2T A +l) (2I..+I } . The total cross section for transition 

to any B state is 


E(2 s ^)tf(S) 

average over spin states. (S takes on 2I a +l values 

the partial cross 

if t a <' J i A V "(2T A +i) if"i A <i a .) Nov a\s) 

section- for various initial values of S, nay be written as a 3um 
of contributions to various possible final spin states, I.e., 

C*sl =X ^% , 

where i denotes a particular final spin 

state of the B + b system. &&\ contains in addition to the dens- 
ity of states in ener~.y, the squared matrix element for the parti- 

cular transition represented by cnSTf . For transitions not 


serving total vector angular momentum , 0(5). 


For examele , 

Ch. ¥111 

Compound Nucleus 


to ■ g i 
Be^ + H 1 


Design an experiment to detect the inverse reaction" 

Li u + He" 

(Design of the aloha particle source will depend on the thresh- 
old energy for the inverse reaction. From Allison, Shapes ard 
Smith, Biy a .Rev. 21 550, or from Hornyak and Lauritsen, Rev" 
Mod. Fry s. SO, 202, we find that Q for the forward react iorTis 
2.115 Mev. In the reverse reaction, in order to get 2.115 Mov 
into the center of mass coordinate system we must give the alpha 
an energy of about 3-5 Mev, and this is the threshold for the" 
inverse reaction (See section A). Design of the Li6 target and 
of the detector, and determining the required aloha beam strength 
require knowing the cross section. This is got by detail balan- 
cing arguments from d^^^ , taking into account a spin fac- 

n° r o° f m 7 ^ - ™ ThiS cr ? ss section is f ound in Livingston and Bethe, 
'-. rev.Mod.Hiys^ 9 245, p. 310, or in the original source, Allen, 

-, ^...^u.iny^ y ^-j p. jiu , or in the original s. 
Ki.ys.Rev ^1 182 Tl937), and is 5 x 10^9 cm 2 " at 0>1 



cross section for the inverse reaction Increases raoidly 'as^the 
volume of phase space available to the proton is increased 
therefore it Is advantageous to use alpha energies an Mev or 
more above the threshold of 3.5 Mev. Higher energy orotons also 
penetrate the Coulomb barrier readily, and are easier to detect 
A qualitative curve of cross- section for the forward reaction 
as^unction g^^is given in Hornyak and lauritsen, Rev. 

D. The Compound Nucleus 

""in the diagrams of section B it was assumed the lH^was 
approximately constant, except for the Coulomb barrier factor 
Often, perhaps in most cases, the matrix element has irregular 
variations. This phenomenon is called resonance . For ^xamol- 

if [n J S J Process 1» indium, there is an extremely pronounced 
peak in rf at a neutron energy of 1.44 e.v. d reaches 27,000 
barns at this energy. (one barn is in '' '" ' " 
this resonance peak is 0.042 e.v, = p 

curve of rf vs. energy has the form 
is the resonance at E R - 5.2 e.v. rf 
for the (n,# ) reaction in silver. 
In this case d reaches 24,000 
barns, and the peak has a half- 


em .; 

l7(E-E R } 2 . 

The half-width of 

the resonance, the 

Another example 


H - 0.063 


consider the reaction n 
for a, 0; assume for A, 
initial spin states : 
tial spin s bates for 
= 4; for S 



+ A — >C + 

±; and for _. 
^2(l)+l)(S(l/2)+l) = 6. 
total angular momentum 3 
1/2, (2(l/2}+l) =2. 

*)s = i < *%-f ) + 

Now Hie first term represents 
states having S = 3/g. For a 

spin3 are, for n, 1/2; 
The total number of 
The number of ini- 
= 3/2 is (2(;5/2}+lJ 


transitions to any of the final spin 
given initial orientation, there is 

only one. Similarly, the second term represents transitions to 
any Imal state having total angular momentum 1/2. hut dnce the 
spin of the a = there are none, so 0T(&) =* 0. Mien orbital ang- 
ular momentum is involved, there may be more than one way in which 
tne given Initial state can form a final state, so that &&) for 

tf^lll' I 3 a T °^er the various possibilities. See BethTand 
Plaezek, Pays. Rev^ bl 450, appendix. 

148 Compound Nucleus Ch. VIII 

The explanation of this phenomenon lg baaed on the assump- 
tion that the transition A + a— hB + b occurs through an inter- 
mediate state C: 

A + a — *- C —- 3 + b 

State C Is the "Compound nucleus". The idea of the compound, nuc- 
leus is due to Bohr.* 

The idea of how resonances in cross section result from this 
assumption can be obtained from the quantum mechanics of second 
order tnansitions. The probability of transition, per unit time, 
is given by "Golden Rule No . l" :** 

trans, urob./sec — 




2 /energy ■ 

x density or 


VIII. 19 

provided there are no direct transitions from A 
section is, from VIII. 9, 

, ' ...:-K 

which becomes , analogously, 



v*v b 



"A^B-t^ I Ea-^ 

% v b 

:o B , The cross 

VIII .9' 

VIII. 20 

Near E^ = E c , (resonance), tS is large. This formula gives infin- 
ite <5 at the resonance energy, hut the formula doeH not take into 
account the short lifetime of the compound state. A correct 
formula is derived In section F. 

The life -time of the compound state is long enough for the 
nucleus C to "forget" how It was formed ,***and this results in a 
basic simplification in the interpretation. 

From the Heisenberg relation AtAE & }4 , the lifetime of the 
compound nucleus and the uncertainty y in Its energy are related 

P > JL 


VIII. 21 

The reasons why the compound nucleus has a. lifetime greater 
than zero are the following: 

1) For charged particle decay, the barrier factor (VIII.12) 
reduces the rate of decay, 

2) Decay by X radiation is very slow compared to the times 
in wnich the nucleus changes Its organization: the lifetime 
against V emission is ~ 10~13 - 10*3.4 sec. The characteristic 
time of the nucleus, I.e., the time for a iiucleon to cross the 
nucleus, is —(size)/(velocity) vug 10-2.3/109, or about 1CT 22 sec. 

3) A particularly important reason is the tendency toward 
equlpartltion of energy in the nucleus. The excess energy due to 
the absorption of the bombarding particle is distributed among 
all the nucleona. It Is rare that there is a fluctuation in 
which a large fraction of the excess energy is on one nucleoli. 

4) Selection rules forbid some modes "of decay. 

* Bohr, Nature 137 344 (1936) 
** Schiff, n. 1967 eq. (29.20) 

a-ture 137 

. P- 1967 
**'* Discussed in Peierl's review article In Reports on the Progress 
In Physios VIII (1940), Fhys. Soc . of London, 1941. 

C ompound ETucl eu B 











S. Examrle of an Unstable Nucle us 

" An examnl c of a nucleus which plays the role of an intermed- 

iate-state compound- nucleus for several well known nuclear reac- 
tions Is Be°. 

The ground state Be decays 


2 He 

as follows: 
■* 110 Xey r » 

The reaction is barely exothermic. The G9i*W exponent for 
decay into a's is low due to low nuclear charge, 
see equation VIII. 57, o. \62 . The theoretical estimate of the 
lifetime Is ICT^sec".** , corresponding to a width of between 
1 and TOO e.v. This time is long compared with the nuclear 
characteristic time of 10" 22 B ec . J hence the width of the level 
is small. 

Information on the excited levels of Ee S can be obtained 
from studv of those nuclear reactions for which Beo is the inter- 
mediate compound nucleus state, such as Li' (p , ~y )Be e — j^2a, 
Li7{p J n)Be7. These reactions are discussed here. The energy 
levels are plotted in FIG. VIII. 1. 

l) a-a scattering. For two Coulomb centers, the total scat- 
tering cross section is oo . ¥e may study the scattering at some 
anr-le'"not near (90° in center of mass system is best). We 
expect peaks in the value of CS when the incident relative energy 
equals the energy of excitation of an excited state. For a s 
scattered on a's", the first such resonance should come at 0.110 
Mev (in center of mass system), corresponding to the Be° ground 
state. This resonance is presumably Ten sharp, a few e.v. wide, 
as mentioned above. It has never been observed experimentally. 

Problem, Discuss the possibility of experimentally observing 
the resonance expected in alpha-helium scattering at an energy 
corresponding to the Be 6 ground state, i.e., 0.110 Mev In the 
center of mass frame. 
(The Coulomb barrier keeps alphas of this energy at least 5 x 

10" 12 cm apart classically, so the effect of nuclear forces is 
probably undetectable. Also the experiment is difficult because 
the range of 200 Kev alohas is so short that it is hard to shoot 
them through an appreciable number of scattering centers and 
detect them. Any' attempt to detect a resonance might- be guided 
"by the experimental procedure of Devons ( Proc .Roy .Soc . A 172 
127 and 559 (1939)), who Investigated alpha-helium scattering 
at higher energies. The theory of a-a scattering and its rela- 
tion to the Be^ nucleus is given In Wheeler, Phys.Rev. 5j? 16 
and 27, (l94l) J 

A second resonance, this one experimentally observed, Is 
at ^3 Kev. The barrier factor is lower at 3 Mev, hence the 
state has shorter lifetime and greater width. The half- width 
is estimated to be 0.8 Mev. 

Further resonances In a-a scattering are so hroac as to be 
scarcely recognizable as resonances. All the resonances men- 
tioned so far" correspond to states of even parity . This is be- 
cause a's obey Bose-Einstein statistics and have symmetric wave 

Hemmendinger; auoted in Seaborg and Perlman table 
Rev. Mo d . Phys ^ 20 585- 
Wheeler, Phy s . Rev . 59 27- 

Isotopes , 

Compound Nucleus 

150 Compound Nucleus ch . VIII 

functions , * The incident a ' s Mil hav e ango lar rsomentuiB 024 
with respect to a target a particle. Therefore states of Be* 
detectable by a sea Iter Inc. IjS helium are even states. 

Not all states of Be 8 are ever;. Odd states of Be 8 c-mot 
decay directly into two a's or into the even Be 8 states mentioned 
Emission- of electromagnetic radiation must occur first, 

" : - i -- -'■ "hange to an ®J| n state by "mech- 

Char.ge of parity 

above , 

an even state c ™t 

anlcal" , i.e., r:on- radiative, interactions 
occurs in emission of photons. 

2) bl (p,K)Be . There is a nrominent and narrow resonance 
at a proton enerrrj of 440 Kev . This Indicates that the life- 
time of the excited Be is long and thus that it is an odd 
state. It decays through the relatively slow process of ~W 
emission to the much lower even Be states. The energy of a 
Ll 7 and a proton separated and at rest is 17.2 Mev higher than 
that of the ground state of Be 8 . Tbe Jf 's given off in decay 
from the excited Be b state produced in the IA?fot, ^ ) neaction 
hav e ener gi e s of 17.5 Mev and as. 14 . 5 Mev , indi eating tf decoy 
to uhe two even states mentioned in the oaragraph on a- a scatter- 

That the excited Be 8 state oroduced in Ll?(p V) is oM 
accords with the follov.ring considerations. The most probable 
ease is for the bombarding proton to be in a high S state which 
is even. The hV is odd, as is suggested by the arguments in the 
following paragraph. Then Li 7 (odd) + oroton (S state) is an odd 
Be state. 

The picture of a nucleus as built up of "shells" of ^rotors 
and neutrons, somewhat like atomic electron shells, su.g-ests that 
Id' in the ground state is odd. 

Suppose the average potential for the nucleors Is a square 
•jell The single particle approximate quantum mechanical solut- 
ion to the problem loads to orbits which ^ay be designated' Is 
lp. Id, etc The Is orbit accomodates 2 neutrons (soins opoosed) 
and two protons (spins opposed); ip accomodates 6 neutrons ana 6 
pi'Otons , etc. ? 


would have the configuration; 
protons: Is 2 lpl 
neutrons : 1 s 2 ipp 
o, o 

or a total configuration: (l8 2 lp; ls 2 lr 2 ) , which has ±\l\=-% and is 
nence an odd state. This model of the nucleus is discussed in 

_ / change s 
'■does not 

x -^-x,y 




y ,Zr-V - 

z, then the ? 

* For two identical particles, parity of the state and symmetry 
functions are simply related: If the wave function"' 
ign when space is inverted by the operation 

ias( odd oaritv. If ? chan Ges 

^even ' does not change 

hen the two particles are interchanged in eosition, then T is 

Operation of inversion of space: Y(r-,,r ) — *■ t 1 (-n ,-rvJ 
Operation of particle interchange :f r!'r,) — *- + flro «T 
But for identical particles, r-^ i^rfL ^ s^fi 
^ £lf-^J - T(r 2 ,r-j), and inversion is equivalent to particle 


Be Com|=ound Nucleus 




IB. 5+ 

7 pwi 



19. ',J 

IS. 1 3 




' ' i". 


R 8 


4)0 *) 


Cu ft e s give cross 
section for the partic- 
ular reaction. 

Numbers l), etc. refer 
to subsections of 
section E . 

(Based on "Energy Levels 
of Light HucleV' bj 
Hornyak and Lauritsen. 
Rev .Mod .Phys. 20, 191 ) 

O of energy 

(Energy in Me v.) 

Be c 

greater detail in section 



Compound Nucleus 
in section K. 

3) Lt 7 (p»p' )Li 7 *. This is similar to 2) except that a 
r rot on is emitted having less energy "than the incident proton, 
leaving Li? in an excited state. The resonance in C is observed 
at a oro'ton energy of —1.05 Mev . Li'* decays by emit tine a ° 
of "bout 0.45 Kev. The 0.45 Mev splitting bet ■-.-sen this excitea 
state and the -round state Li? may be due to enermy difference 
between Pl /n and py states of Ll 7 , on the nuclear shell model: 

SS.T3 Mev. 


U t7 





Cem^»>»4 Nucleus 

^r^^Msta*! tsf eder-g/^ \ Mev 

Li 7 + p(l Key) — Be — 

lf£nevg/ = O at, 
of Be s , 


d stat 





+ P' 

4) Li 7 (p,a)He 
> 7 i 


odd, the incident nroton must be in an odd. state, 
state with respect to the Li 7 nucleus. Ko reson- 

Since two a's are in an even state, 

since Li 1 

probably a p' ... 

ahc'es are observed. None is to be expected, since all even 

resonances are extremely broad. Note that all the observed 

re sonance s 7 are odd^level s , ,„__,+. __ D .^„„ r .„ 

5) Li'(o,n)Bn'. A resonance is observed at a proton energy 
of— 2.22 Mev, corresponding to an odd Be b state 19.15 Mev above 
the ground state . ,- 

6) Li 6 (d,a)He 4 No resonances. Evidently Li D + & form an 
even, state, and quickly decay to two a's. 

Problem: Design an experiment to observe the famous 440 Key 
resonance in the reaction Li? + p— ^Be" -+/+ Be 1 - ->2a ■ 
(This exoeriment has beer performed by ha Ike r and McDaniel^ 
using a gamma ray spectrometer which measures the energy of 
pair's orodueed by the gamma ray ( Phys.Rev . ^4 515) and by 
Delsasso, Fowler" and Laurit sen ( Phys.Rev. 5JL (1937)) using 
a cloud chamber to detect the gammas by means of electrons pro- 
duced in the chamber by pair production and Gompton collisions. 
Recent electrostatic accelerators have been equipped with 
electrostatic velocity selectors which provide an energy spread 
in the motor, beam of less than 300 e.v. at 1 Mev. In order to 
take advantage of this narrow energy range, very thin targets 
must be used. These oroblems are discussed in 'Gamma-Radiation 
from Excited States of Light Nuclei" by Fooler, LaOTitaen and 
Lauritsen, Rev. Mod. Hays. 20 236 (1948)). 

F. Quantitative Development of Resonance Theor y; Brelt-Y.'i.-.ner 

Formula . t ^ \ 

In this discussion we use as an example the In, ) reaction, 
1 . c . , rad 1 a t i v e c a otu r e of neu fc.rone , wh i ch is a n 1 mpo rt a nt re;:, c - 

As in the preceding sections, the energy levels of the ini- 
tial and final states form a continuous distribution. The exoer- 
ment picks out the particular initial state. 

^y ie resonance phenomenon that we wish to describe is very 
enerry -sensitive. Ve shall attribute it to the existence of: 
compound rue "Lens state C, connecting with the Initial and fir 
states by matrix elements "H Bt . and Mfat , where the notation is 




Ch. VIII Resonance Theory 155 

given in FIG. VIII .2 We shall neglect matrix elements of the form 
H & q> j Mty&tiA M a y , which connect the initial state with itself, 
the final state with itself, and the initial state with the final 
state without the intermediate G state; i.e., no direct transitions 

from A to B, 

STM"E A CornswJD Nucleus C STATE B 
A+/n C B *t 

inde* a. 

— — index b 

(discrete) (fort-hrwous} 

PIG. VI II. 2 

The problem of computing transition probabilities can be 
done by using the complete machinery of time dependent pertur- 
bation theory-. In summary, if u* are the time independent elgen- 
functlons of the unperturbed states having energies E. , the true 
eigenfunction for the perturbed state is 

Y =1 % A, 



VIII. 22 

where a 1 are the amplitudes of the unperturbed states In this 
expansion. For the method to be useful, the true eigenfunctlon 
must not differ greatly from one of the unperturbed eigenf unctions . 
If there Is no perturbation, a n = for all n. With the oertur- 
bation,^ , the a's change according to the equation 

a = 

^Z H^ 






VIII. 23 

If j just before the perturbation is applied, the svstem Is in a 
state represented by uv, i.e., a, = t- k , then some 'time after- 
ward there Is some probability or finding the svstem in states 
other than the k zh . For states differing in energy greatly from 
state nnm.ber k, the exponential in the above equation oscillates 
rapidly and the change In a ji tends to average to 6. 

to the 
, of B 

states by eL 
E„ by d c (see 

state Is near enough to be important 
the one- level Breit-V.'igner formula.) 
the amplitudes are " 


present problem, we denote amplitudes of A 
states by a-v, , and of the compound nucleus at 
rd ■:/'- ill. 2) . Assume that only one intermediate 

(The result is theft called 
The differential equations 


= "^K 

q, = -^H tc e 



1 a. 




_al A state , 


ct n 

= 1, 

a„ = for a 

£ a 









exoerlment . 

VIII. 24 

At t 



m, Ch. VIII 

Resonance Theory 

154 - ■ n h,rt a ^ a increases, representing 

At t = a a = a. = but a ^ O .a intermediate 

the build up of probability that tne eys airplitu des a c and 

State C. When a c becomes larger tnan , the *»pJ-l de £ ot9B 

a b begin to Increase. Tne a^ 1 *^^"*^ a^ituds that 

A + ^ ri^ous ,ay to solve -r the transition P^abllltles * 
is to carry through the solution of the system oi 

^^ollO^E 1. a practical way to K et the answer quickly. 

Assume for the sa.e of «*^^ t t*^ ^tn^ 

compound nucleus state is ^^^^^^Son^rooabillties are 
tegrate either to A or to B. The transit^ P ^ ^ ^ 

^Sa b Ii;SiS B o?S^ta?e SSst'a certain mode of decay. 

Probability of transition to B - ^ - £ [Ate 
(per sec . ) 

= i4^4 c3 VIII. 25 

where we have put ^/c = p y , v y - B ■ 

Probability of transition to A = ^r -^l^vi * 
(oer sec . ) 
(Trc Wave functions are realised in a volume Jl = 1.) The decay 
from the state C ivould folio,; the eauation; 

Prob. of occuoation of state G - g 


VIII. 27 

Actually the probability of occupation of a states ^am P litude| 2 . 
.If we ignore the phase factor, _vrejwrite_ 

amplitude = NJprobability 
t/ I ±A_\ 



5 4 — ( 

VIII. 28 

Kere thorou^ treatment confirms the result of this step. 


VIII. 29 

VI 1 1. 30 

n ^P + P -fe/a) times the total probability of ^structlon 
I rrU+ U -^% % ^l intermediate state, per unit tune. 

( r is sometimes defined ^h^T^T^ZlA^ ce^ritSn 
probability of decay p er uoit -.lme.J me.- vjl-l-u- __ . 

* Breit and W^r, BHfeBg* g g9 Clg6) 

** Froir now on, Q c is not ae^essaxxiv e.quaj 


Re s o na rxe Th eo r v 

' v 155 

We muBt add a tenm to account lor the accretion 

J*., e - —*■ ""** s* l.^-x:u uu account lor the pncrp+in 

of the state G. State C is IiM frcm a+ .^ . h ' „ C f^f 

appro-n^iate +pt«, fv,™ -,-. t L IT irom state A accordin- to the 

ViilJJ, rametf £* " S ati0r ;r e °^ **»*«* to equation 

we try 

VIII. 31 

SmK^SitV^S S^TS^f eqUati ° n 3atlsf nn G the initial 

a, == 

^H^te^-^-e* 1 ? 

VIII. 32 

tJsl nr fho ,,-„ ■*-•„■ experimental 

wid?L! h t\rSiSS£? ?i^ C Sr S t^ °^ /tol ^e Of the energy 
turns out usually to be ?esf ThS 1 gSfg ™ 3 St f e G ' ^ ■ „ 

seconds the term exr>( Vi /%) IT ^ C * Arter ^™ i^ 13 

Jn e;x i'\-\ V)iJ is nearly 0. Therefore 

VIII. 33 


ri2. 8t Sen^ lldS ^ ^ ^ am ° Unt re3?rea9n ted b ? **• «1« of 

Number of reactions to the ri^t/sec 

B + y } 

= Ucl "^r 

Number of reactions to th= L6£t/fie* 

(A + n^G-^A + n 1 ) " = 

L X 

VIII. 34 



examine" k] ^"it^SarS^^T^, ^ ln the flml fo ™> « 
c „ J ' cl early Ilas the form of a resonance curv^- 

By definition, T aaJ T are fcrj* ■•'-..-. 

connected by 2^ = ^. Slne:# " 
I I turns out to be approx- 
imately the uncertainty* in 
ener G y of the state, this "is 
an aspect of the Heisenber- 
uncertainty relation AETS'tf. 

"Energy /v.dt h eft 
%_ maximum ^ O f -1 

Returnin G to equations VIII .34, ;,rit!n. ; them explicit! 

Wo. ncactions/seo foWl* 

A^G^E (per = fT~ . — 

unit beam density) ' ^v t e~-£«J 

No. reactions/sec |>t co '* 


unTt bTam density) ^'+ (£t- E g J a fcjj 7 ~ *W)^<ft 

Viii. 35 

where the definition of Cf 1 r equation VTTT 7 i„ 

assume the r,robl«n jq r„,™*n , q \ , lon ' m '7 is used, and we 

cf neutrons"= lT -iodized to volume^ = 1 and density 

156 Resonance Theory Ch. VIII 

(in deriving this result more carefully, one should substitute 
a c into expressions for a a and a, , then put these g, ' s and a^' s 

into the correction expression f on a , and check that the result 
is consistent.") , ,-; 

J _ ^ ] ^ Ca > { 
M ^j p, + { ^ T VIII. 36 

21; i>u„r 

The next step is to replace \%J with more useful parameters. We 
use equation VIII. 26 which nay he written 

M 2 "" ""'" 


* = i¥?T 3 1 ^ ! v 


VIII. 26' 

:f | ra denotes U for v„ = velocity of neutron if it had the right 

energy to hit the resonance exactly = v R , then 

~i^)H*J e M' 

VIII .37 

provided 1Hcq,| varies little with change In velocity. Rewriting 
VIII. 36 in terms of the parameters v R and ["*- , 

°fm,*l "- 

C%/n) — 

2g 1 2ir^rJ 

VIII. 38 

VIII. 39 


Wow use the relation ^ =i U, K Cv/v R ) , where v is, as always, the 
actual velocity of the neutron. Also,^ ^ tf/Mv , | ^ ^ /Mv R . 
M = mass of nou.tron . 

rvr R 

VIII. 41 

v = actual velocity, v R = velocity of neutron to hit resonance 
exactly. .The aensral nature of VI I I. 40 and VI I I. 41 is shown in 
FIG. VIII. 3. Note that for 
energies "below the resonance , 
the eqvntioas above behave as 
described in section B, p. (44, 

So far, angular mom°ntnm 
multiplicity has been neglected. 
For the .{v., IS ) reaction the spins 
are: feutron, I H = l/2 

Initial Nucleus, I A 
Compound State Nucleus, I c 

-' J C^ih/h) 


CJli VI11 Resonance Theory 

Final Nucleus, I B 

Photon of fixed momentum has two degrees ef 
freedom and ia lifce a uarticle of sein l/2 
(see ref., footnote *#», p,|^j ' 

---^j* Sl °'" r r" eutr,0n tactions the neutron usually has no on-it-i 

DJmTl X G - % i 1/2 resonate. The cross sec- 
tion must be multiplied by the probability that, -or giver, I. am 

are^Tf^^/S ^ ' *** "^^ ^ the ri " ht ***»t*tl<m. There 

- T Wo V + * initlal St " tSS havl ^ ; t0tal an C^r momentum 

- I A ^ + 1/2, ( 3pln . a parallel), and 2(l A - i/o ) + x st ,, tR3 ^ 

total arxcular momentum = I A - i/ 2 , ( api , s ant inarain el ) . Ihfi "" 

total number of states Is a<l + i/ 2 } +1 + 2(l A -i) + i = 4l + ^ 

The probability that an ineomlm neutw™ ,diT ^ f ■ 

state uith the nuclei-. j^ mlIJ S n "' ltl011 ■'HI f™ a scin-parallel 

4r a + 2 

21, + 1 

VI II. 42 

and that the neutron forms a state with spins anti-parallel is 
j^I A -/ a ) + [ _ T A 

4I A + Z 

23* + 

VI II. 43 

V: ' '' " - l be ^oniblned, ~ivin S for the final equations; 

VI 1 1. 44 


Cfyyft,} — 4-TT A K 



v s +^y 

21* + 

VIII. 45 

where the + is used if I Q - r^ + w £ - ar)d _ 

There is little or no resonance if b i t 
neutrons. C F ^ 

if I 

1/2 , for 

- 1/2 


in mo-°o 1 S 4r + 10n CU ^ eS r ° r £ ey -*^ ^bsorntior and scatWrinr 

il . , "Science and Eiipi- 

m^j^ 19259 fl.947) or in Goodm 
neerm;;; of luelear Power", Vol. I. 

^ m „ At the ^ddle of the periodic table, A = i 00 to 1 50 fn ¥ 1 
aroc esses are ^rmil npn+ n-p+ D v, «„ -. " -o<J , in, C ) 

for /Sta is ~ fSr tL^ pGrl f iC f^' the M ^0 energy 
Shis amount At +m * h ' L compound, nucleus is excited by about 
closed s^c^+niL ? r r °f ^^ation, the levels must" be 
VIII 4 3 ht t ° n " U ? 5? r ^ c -- 0Sel 7 spaced resonances, FIG 
viii ^. But at Iovj excitations vre find that the ener-r levels 

***** Cr ° 3S ^^^^^^^^u^Ti^rcso process^ 


Experimental Resonance Data 


are of the order of 0.1 Mev aoart.* Evidently the— Is a -reat 
S^ Se ^^ 1SVel dens "y wit* increase in excitation 

onei {^y . 

The increase In level density has been lucidl^ eXDlaiaed by 
Boar. Suppose that the nucleus resembles somewhat a collection 
of narmonic oscillators of different frequencies «* . Eacl has 
energy gc**^ (ignoring zero point energy ). The total energy is 
E =1W^ W ^*-~^u^ . For low OMltatior _ f fBW Qf the 
Kj differ from 0, and the energy channea in iumns of ~&*J which 
we may take as -v, 0.1 Kev . For large excitation, say 10? e!v 
many rjj s are largo; there are in general manv sets of m*a that 

£l> a J^S 1 e r- Gr ^ near t0 a lar Ce value of excitation Each 
such set of n t 's will in general giy. e a slightly different total 
energy, therefore the density of levels will be large. This result 
clearly depends on having many nucleons. Ve expect few energy 
levels and few low energy resonances in the light elements the 
number of resonances Increasing with A. This expectation, is fnl- 
lll.lea to some extent, because there are few loiv'e-iem (<1000 
e . v . ) r e s o na n c e s In el erne nt s lighter than mangane s e . ^ ' " 

+ n ( F S r , vo ^ hi #i atomic number, the density of levels accessible 
to [n^ ) processes does not increase; altho hi~h«r A means ior P 
degrees of freedom and denser levels, «**«« more 

the binding energy of the neutron 
decreases from ~ 8 Kev to 5 or 6 
Mev, and thus the states for for- 
mation of compound nuclei by absorp- 
tion of a neutron are at lower 
excitation energy and the density 
of levels is correspondingly less. 
These opposing trends tend to can- 
cel . 

Not all levels are detectable 
by (r.,^ ) processes. Those for 
which the spin of the compound 
nucleus state is not equal to 
■^A ~ 1//2 ' a "l/oi" parity Is not 
conserved, do not give rise to res- 
onances in { n> X ) cross section. 

Throughout the periodic table 
I Y is roughly constant at a value 
0.1 e.v. v.dth variation by a factor 
of 10 either way. 

For medium and heavy elements, V n s= lcr 3 
ponds to a life time against neutron decay oi ■ 

Is large compared to the nuclear oeriods of about HT"**- sec- the 
captured neutron moves around — Iff? times before a neutron escsnes 
ihis very low probability of escape can be understood by consider-" 
lng partition of energy among the many nucleons. (See n". 14 & ) this 
process to make neutron emission slow operates bettor as- A becomes 

In light nuclei, VL. is larger than in heavy nuclei, as ore 
expects from the partition of energy argument. " \Z ler-e means 
large probability of scattering comoared to cantnre. In mangan- 
ese^ for example, the resonance at 200 e.v. is" almost pure scat- 


Levels of 


c o rre s- 

e . v . Shi i 

10~ J - seconds, this 

* Known from gamma emission spectra and from comolex 

alpha spectna. 

lM e v 

c; the 

escapes , 
}. ) Thl s 



Nuclear Gas Model 

out. 1% as larpe as the 
VI I I. 44 

teri-'i" * Trio rjeslc in absorption is only . 

scattering peak. The Breit-Ki -ner formulas VI II. 44 ano 05 are 

fitted to this resonance by putting b> &*1 o.v. ana V^ 10 B.-.V.. 

In TP rv UPht elements, there are no (n, 5 } re sor-.ar.ee s at 
ell. There are, however, (n,p) and (n,a) resonances. 

The Breit-Wi-ner theory may be extended to reactions involv- 
ifin clia^fed rarticles by ircludinp the barrier factor clue bo 
Coulomb" forces.** For encrpies above the hei-ht of the Coulomb 
barrier resonances are observed. For example, the reaction 
''■' ^ H? has a 'uroad resonance at 0.27 Kev. 

Li 6 + r_ 


Problem. Assuming that U»\y, find the possible values for 
C for the 280 e.v. resonar.ee in manganese. . CJ - , , \ 

( i at 280 e.v. is 2.73 x 10" -1 cm. The spin of hn JJ is 5/2 J 
Froi? enuation VIII. 45 f 

At resonance k = 0. Then ipnorinp Vy compared to k , we pet 
If v;c hnov: that P^lOO hV , ve can also find %,19. 


Stati sti cal G-as Model of the Nucleus 

Va r i ou s m od el s o f the nu c 1 eu s empiia size various^ different 

features of the nucleus. Me si nple simple f" -}} 

nuclear properties. V;e shall consider tne statistical oi ^f 
model, then' the liquid drop model apolied to ussion, anc finally 
the nuclear shell model,*** 

Tne -as nodel nic tares the nucleus as a C as of protons and 
neutrons/' This model ignore, surface effects Itte na .piliaritv, 
a serious omission. The volume of the pas Is W ,5 ; "^ 
Due to this restriction to a small vo u,e ^^ILsTv^r/Sfih 
mm for s rertide has widely spaced levels.*--" Dffles.s very p^gn 
e^ciStlen^nercies are postulated , P^I^a will ? c W toel jx- 
_.« + w911 »i,i 6 states. For the usual nuclear excitation encr.ieb 
oflQ lev or so! the nucleon ras Is almost completely aepenerate . 

We shall use the m model to compute an approximate nuclear 
.etentiarSii'depth, !o explain semi^antitatlvely the increase 
in nuclear level density uitn ener G y, and to consider smisBltti 
particles as an evaporation process. 

The excitation of the nucleus, the ^tert to .chich hipiher 

T - cerr-sS? to complete degeneracy . At T = tne number 
L-%?aKa ri to the hiphest one occupied just equals tne nun ner 
or reticles, either Z or A-Z, depending on uhether the proton or 
neutron part of the &s is in question. 

71 6 


* Seidl, Karris and Lanpsdorf, Phvs.Rev 

#* See Bethc- B, Ch. XIII, pape 186 . „ fl ... fa1/1 

*#* Various nuclear models are discussed by L. Rosonleiu, 

Nuclear Forces I I, n. 185 , and by Seine B, p. (J 
**** Equation VIII -5, "for enerpy density of states, cortains 

the volume . 


Nuclear Gas Model 


The number, n, of states of momentum less than r> ,„ of a 
proton confined to a voiumeJQ, is 

r ' = 2 ■ — ; rvj (factor 2 is for snln, see p. 142} 

3(2ti40 3 

At complete degeneracy n — Z, therefore 


* f 

%Z ■-■■ (? v )^k& 


VIII. 46 


VIII ,47 

In the crude approximation, number of neutrons — number of protons 

4L - (3Tl) * inm 


= 1.05" x l& cgs umts 
0.5*^A 7 3 ' : VIII < 48 

;S!: = :S independent of A. The corresponding kinetic energy is «v 21 
Mev ^y^a . ^Mf, Is the kinetic energy of the highest occupied 
neutron state. This energy is measured from the bottom of the 
potential well. Further more,yW is about 8 Mev below 0, a 3 shown 
in FIG-. VIII. 5 This fixes the depth of the well for neutrons. 
Actually there are fewer protons than neutrons in the nucleus, but 
the topmost proton level must have energy^^o , otherwise there 
would be 3 decay. Therefore the depth of the well for protons is 
somewhat less than for neutrons, by this model. Also, the Coulomb 
potential acts on protons . 

Ipl-oton |ootenilQ I v. 

Oeotron potential 

8 Mev 

~ 21 Mev 


The following, is a calculation,,! corresponding to a usual 

value of nuclear excitation. From the statistical mechanics of 
a degenerate Fermi-Dirac neutron gas.* the total energy of excita- 
tion (measured above the T = level) is, for neutrons 

(FIG. VIU.&) 

# Mayer and Mayer., Statistical Mechanics , p. 374. The total energy 
measured from the of kinetic energy is 

1 ; + ""J VIII. 5P 

VIII. 49 


<m. viii 

Nuclear Gas Model 


How put In Tor / M 6 = p^ ax /2M , and then for p put VI II. 47. The 

result is 



ftfejA h&^s 

rV^^^-^^f) 2 





p = fW ^ f^f f/rf! 

VIII. 51 

VIII. 52 

where ^ =nuclear volume, T = nuclear temperature, and M = mass 
of a nucleon. The total (neutron plus proton) excitation energy. 




E t - E n 

+ E 





VIII. 53 

Where E^ and (kT) are in Mev. If 
A = lOO^and Z - 44 , HL = 11 (kT) 2 . 
Then (kT) = 1. Kev corresponds to 
Ex— 11 Mev. If we attribute to each 
kT a "decree of freedom" in analogy 
to classical statistics, then in this 
case there are only 11 decrees of 
freedom, whereas there would be 3A 
degrees of freedom in the classical 
E£ s . 

The increase of energy level density with energy of excitation 
can be computed using the statistical-mechanical definition of 

Entropy = S(T) = k(ln w(T) - In w(0)) VIII. 54 

where w is the total number of quantum states available to the 
system at the specified temperature. Fro is thermodynamics, 

str)= f T ^ » r T 2aVTdT = Zo ^J sine* E-a^feT}' VIII. 55 

a a 

So 3 =^ ^fl^^ r 

From the definition of entropy, 

,7%. _ _ US (Tj _ dens ity of states at temo, 
uj ( 0) ~ density of states at temp. 


VIII. 56 

e = e 

11 (kT)^, 

For A = 100 and Z — 44 (same example as above), E 
and at E ~ Mev, the excitation of a compound nucleus after 
capturing'?" neutron, e ^ ^ e l9 ~ po 8 . Therefore, if near excita- 
tion the level spacing, is ~ 100 Kev , at 8 Mev excitation it is 
~ 1 millivolt, by'this'very approximate calculation. This spac- 
ing is probably much too small, although one expects this calcu- 
lation to give' a greater level density than observed by resonances 
because resonance experiments detect only levels with spin compat- 

ijgg Nuclear Evaporation ch ■ "Hi 

ible with those of the initial particles. 

Nuclear evaporation . The emission of a neutron from a nucleus 
may bo considered an evaporation of a particle from a statistical 
group of particles held in a potential well.* In such an evapor- 
ation, the particle carries away an energy of order of magnitude 
ST, which is, in general, much less than the total excitation 
energy of the group of particles.**-, For example, suppose a nuc- 
leus'" of A = 100, Z"= 44 has excitation energy, £=20 Mev. The bind- 
ing energy for one neutron is about 8 Mev. The temperature will 
be about, ""l. 3 Mev. A neutron will, on the average, have kinetic 
energy of about 2.6 Mev after escaping. Therefore the nucleus is 
left With an excitation of about 10 Mev., which is sufficient to 
emit another neutron. After the excitation is reduced below the 
binding energy of a neutron, the nucleus may decay dj gamma omi-s- 
si on. 

Protons encounter the Coulomb barrier, which has a height of 
^j .gr-10 Mev for medium weight elements. The probability that 2 + 
~ 5 Mev is concentrated on one proton is small if the total ex- 
citation energy is of the order of 20 Mev, therefore protons are 
less likely to' escape. At small Z, the Coulomb barrier is rela- 
tivelv small, and p emission processes compete with n emission 
processes. For largo Z, n emission dominates. 

When the energy delivered to a nucleus is very 
than the binding energy of a particle, as in nuclei 
c o sm i c ray part 1 c 1 e s , a ' s a nd ev en larger nu cl ea r 
"evaporated". In photographic emulsion or in a cloud chamber. 
a "star", having 3 to — 20 or so prongs, is observed r (See 

much larger 
excited by 
frarj-ients are 



Plot the probable number of neutrons that an excited 
nucleus will emit, as a function of excitation energy. The 
plot should take the form: probable 

Plot up to energies such 
that four neutrons have 
probably evaporated . Discuss 
the probability for emission 
of a proton at this excitation, 

nurvber <£ 

Assume A = 130, Z - 54, and that the binding energy of a ncutron 

or of a proton is fl Mev . 

(The temperature of the nucleus changes only a few percent when 

a neutron leaves, so the error is not great in assum'ng that T 

is constant. Gamma decay may be assumed negligible for energies 

of excitation "of over a few Kgv above 8 Mev (see Bethe B, 

p. 160). Considering emission of mavtnons first: For the energy 

region 0-8 Mev, no neutron can lie emitter]. For the region 

{3+n, f sv: Kev)to 16 Met'-, one neutron is emitted-. For 16-24 Mev 

there are pb.mpeting modes of decay, namely, l) one neutron may 

take cnoumh energv'to prohibit further evaporation, 2) two neut- 

state population 

met ion for the nuc- 

Energy, E b 

ro"-s may leave. The 

lcar mas has the f orm : 

We may approximate the 

tail of this function b 

an ezepo .ential, and say 

that t.-~e A rrobabillty for 

escape = C-v/u.-tt'e'^' 1 '; £>Efe 

where C is a norrmlisatlo:: 

factor and £. is energy measured atove^, , the Fermi energy, th e 

Frenkel , Phys .Zo it, s .Sow .1etu.nl on 9 533 (1936) (in English) 
■- Accord in" to Weisskopf, Phys.Eev. ^2 295, the average energy 
of evaporated neutron is about 2kT . 



Nuclear Evaporation 


energy of the highest occupied state at T - Let i = E - 16 ■ 
^, 1G the e xclt -tion ener^. Then the probability that oilfe neutron 
takes out so much energy that leas than 8 Mcv is left ii " ' "" 

> - */VT 


c" d et e rsai ned 


8 + 5 

c At 


= c £, kT where 

setting P, =r 1 at 6 

0; C" 

p l = 

- p -feT 

Probable number evaporated, excitation 8-16 M°v = P + °P = 
P L + £(1-P 1 ) =.2 - e"^ T 1 

For total, excitation 24-32 Mev , there are three eomnetinr 
process: 1) one neutron loaves not enough energy for further 
evaporation. P-, = e"^ 8 *»% calculation^'slmilar" to above . This 
is neplipibl.e. 2) the first neutron evaoorated leaves enough 
energy for just one further emitted neutron. The nrobabllttv 
for t'nis is i-tt+s .£. _ S - - - ./ 

p ? - G l e FT i £ = e * 

|) the. first, neutron leaves enourh enerpy for two more neutrons, 
ana either one or tv:o more are omitted. The probability that 
case 3 J occurs vrith just one neutron beinp evaporated subsequen- 
tly can he shown to be _„ g / - 1- \ 

p 2 = .Ff(, e ' V- Then the Probability 

for ease 3 J with- ftwo subsequent neutrons, i.e., three altogether 
is ■ j ■ 

% = 1. - Po = 1 - H - P3 = 1 - 

The average 

= 3 - e 

number for 


the 24-32 Mev rann.e is 

The plot turns out as -iven: 
For protons. the barrier is 
effectively higher, by an 
amount that Is mnroximatoly 
equal to 0,9 times the peck" 
height of the Coulomb poten- 
tial. T> Le probability for 
penetration of a Coulomb barr 
by a charged particle is 

P = gl^PT^^'K^ 


infi W)=i 


where x = Enerry/Barrier h&i^hi 
(after Bethe, B*, p. 167) 
A, and 2 1 are mass and. charge of particle, Z ? is charge giving 
rise to the barrier. A 2 , the atomic weip-ht, enters through the 
nuclear r ? ,n;, n . ft -. ™ + „ • Curves of probab- 

nuclear radius, R^. For a proton, A-.Z-, = 1 

ility of oenetration for three valui 



\\ A = 


\A 2 -a 

2 \ 

X \ 

A, for protons, are 

rVotnri rVstnstod/eftcy 


'Eme rgy 
« = Barrier 




The ratio of probabilities for emission of a neutron and. of a 
proton is ._. ->■ ?wev 






~ .00 37 

where 9 Mev - 0.9 of the 10 Mev Gculomb barrier. 1.65 is kT 
c o r re sp o nd ing to an excitation of 38 Kev , av e ra ge numb e r of 
neutrons emitted = 4. 


Fission ** 

The most useful model for explaining fission phenomena Is 
the liquid drop model (see chapter 1,0, p. <S ). This model per- 
mits calculation of the change in potential energy when the nuc- 
lear drop suffers an ellipsoidal deformation from spherical shape. 
If the potential energy Increases, spherical shape is a stable 

The two contributions to the potential energy are l) 
capillary energy, 2) electrostatic energy. We wl.ll calculate 
the change in these contributions to the potential energy for 
a constant volume prolate ellipsoidal deformation given by the 

ma lor semi-axis = a = R0+ e ) 


minor semi- axis 

b = R/\fi 

1 +£ 

O— <3P> VI11 - 58 

where R = Initial radius, £ = parameter giving extent of deform- 
ation. (3-S- approaches the square of the eccentricity of the 
elliptical section as both arra roach 0.) Volume is invariant: 
V = "(W3)ab 2 _ (4n/3)R3. 

l) The capillary energy is proportional to the surface area. 

Ellipsoidal surface = 4tt P 2 ( 1 + 2/5 £*+...) VIII. 59 

The capillary energy was c omp.it ed In Ch . I , p . 7 , and found to 
,;De 0.014 A^ for an unexcited (spherical) nucleus, therefore 


merry = 0.014 A 3 (l + 2/5 E*+...) (mass units) 

VIII. 60 

-, n a 

2) The electrostatic energy = (3/5) fe-Z^/R) (l - 1/5 %hk &t 
sphericity the energy is, from Ch . I , p . S , 0.00062 7 Z-/A, 

Electrostatic energy - 0.000627(Z'-/jml - l/g E ) VIH.61 

This is evidently maximum at spherici" 

The total change is 

£ (2/5 x 0.014 A 3 - 1/5 0.000627 &'/$%) 


Spherical shaoe is stable if this is +; unstable, if -. Roughly, 
■electrical energy is proportional to A* 3 , capillary energy to A ^, 
Therefore the electrostatic energy term dominates at high A. The 
expression VIII, 62 gives a criterion for stability for given Z 

and A, namely, 

Cvt-'i - 

Spherical nuclear drop is stable if t "C4.7 VIII. 63 

* Bohr and Wheeler, Phys.Rev. 56 426 (1939); Frenhel , J.oi 

Physics, A^ad.S c l.U.S.S.R. Vol. 1 Jo.2 (1939) (in English) 
**Gencral reference: W-.E .Stephen, "Nuclear Fission and Atomic 
Pov, r er," Lancaster, 1949. 

Oh. VIII Fission 

The plot, of Z 2 ,/A shov:s that elements up to U 2 3S 
stable by t]iis criterion; they are In reality. 

Although spherical shape 
of the nuclear drop may be a 
relative minimum of potential -2 
energy (i.e., metastasis), the ^ 
potential energy may be even 
louer for completely" separated 
halves of the drop. To investi- 
gate this, v:e can compute the 
enerry of tuo separated spherical 
drops of equal volume having a 
total volume equal to that of the 
combined sphere, and compare this 
ulth the energy of the original 
d ro p . The ma s s f o rmu la 1.8, 
p, 7 , may be used for this ■■ur- 
pose. If M(Z,A) denotes the mass 
are Interested in the difference 


at least, are 



— - - 








D . 






of combined nucleus, ther: ue 


HV 2'1 J 


0.014 A 5 (l-2 3 ) + 0.000627 



This gives the 
volurr. e fragment a 

Ccrence In energy I or 
At, closer distances, 

Infinitely separated half- 
still not" touching each 

In FIG. VIII. 7, the 

other, the 'potential increases due to Coulomb repulsion. These 

facts permit d raving an approximate potential energy curve. For 

y236 the separated halves have energy -169 Mev relative to the 

combined non- excited nucleus, FIG. VI II. 7. 

early part of the curve 

represents deformation 

of the sphere into a 

prolate ellipsoid of 

small eccentricity, and 

Is known from equation 

Fill .62; it la quadratic 

in the parameter £ . For 

Fully seoarated fragments , 

" ~ no"' 

the curve Is 

I) r 

it to r c , the separation 

CoulomD betential 

Distance j mengm^ 
ith E. at sb^Vi t d^Tov-^dtiDn 

distance at which the tv.'o 
half-volume fragments just 
touch, the potential dev- 
iates from the Coulomb lau 
due to the onset of nuclear 
forces. (in the case of 

-,f- ^Volume Spheres 

\S- J0 , if the Coulomb lav; just touching pi S .y[if. f 7 

held right un to r , and If the fragments remained strictly spher- 
ical up "to this point, the potential energy would be 210 Mev J 
The c v rv e in the re pi on near r is c ompl 1 c a t ed , Cal c : : 1 a t i o n s o f 
potential energy for large deformations (up to about point B on 
FIG. VI 1 1. 7) of the nucleus have been performed by Fran!-: el and 
Metropolis ( Phus.Rev. 72 914 (194?)) using the "Enlac" computer. 

Due to aero point energy characteristic of 
systems, the system has energy slightly above t" 
potential well . 

There is some chance that 
undergo " snor-tar.eous" fission 

the nucleus will 
:y tunneling 

qu a ntu m- m e c ha 1 , n I c a 1 
■ e bottom of the 

fig, vm,7 




through the barrier. The probability is lovr, because the masses 
are large and the system approaches being classical so far as 
tunneling through barriers is concerned. Therefore, effectively, 
there is a threshold excitation for fission. 

In 1939 Bohr ana V.heeler* deduced a rule that has been 
verified experimentally. It is that the reaction (n,f) 1b pro- 
duced In elements haying an odd number of neutrons oy thermal 
neutrons, but with elements having an even number of neutrons, 
fission Is induced only by high energy neutrons. This is because 
changing from an odd number of neutrons to an even number of neut- 
rons releases one or two Mev . This rude agrees with the table 

of which is reeroduced here. The 


Possible Bl'vichviq £^e.v 







- 7 
~ 5" 


given In Ch. I, p. 7 
rule Is exemplified by 

fr 35 and U 238 . The bind- 
ing energy of a neutron 
to the former may be some- 
thing like 7 Kev; to the 
1 a 1 1 er , wh i c h already has 
an ev e n numb e r of neutrons , 
only about 5 Mev. Fission 
induced by thermal neut- 
rons occurs in Ij^3?5 o U t 
not in 02^8 _ Xli tnis case, the barrier to fission is presumably 
around 6 Mev. in height. 

Asymmetry of Fission Fr , _ 
to split into equal fragments 


sent distribution. Nuclei tend not 

Fragments tend to cluster in two 
zones, as shown in FIG-. VIII .8 and 9'. Referring to JFIG-.VIII .3 , 
the fragments, after formation, 

move toward the line of max- 
imum stability hy p decay. 
There is no adequate theory 
for this a symme t ry . 

Extensive data or nuc- 
lear species produced in fis- 
sion appeared In Rev ,yod . P^s , 
18 513" (1946). 

fission fragments 
tend to be in these 
regions, rather than 
midway . 

Stable nuclei 
— i 

FIG-. VIII, 8 






lo" 1 


/ 140 N^ 






1 0<j 1£B 




Kejnarhs on Fission Fragments . Fission fragments nrroduce heavy 
cloud chamber tracks about 2 cm. long in air at NTP. * In contrast 
to ionization produced by a particles, the density of ionization 
here decreases with d 1 st anee t rav e 1 1 eel , as sho we i n FIG- . V 1 1 1 . 10 . 
The explanation is that the fragment , Initially to a large extent 
stripped of electrons, gains electrons as it slows down. Its 

* Bohr and bheeler, Rhys. Rev. 56 426 (1939) 

eh. viii 

Nuclear Orbit Model 


■■fission fragment 

About 1% 

effective Z decreases, and ionization depends on Z (equation 
11.10, 1. *bh (The a picks up char G e also, of course, but the 
effect Is smaller.) 

Emission of Neutrons in Fission. Immediately after fission, 
when the f ramnent" Tr drops' r are some- 
what like: OQ ' each fi*^>»ttb 
uossessss considerable excitation 
energy. This energy la used to a 
large, extent to evaporate neutrons 
There is approximately ono neutron 
produced per fragment 

Delaved Neutrons 
of the neutrons emitted by fission 
fragments are emitted at relatively 
Ion": times after fission, I.e., 
from 55 seconds to a fraction of 
a second. The explanation Is as^ 
f n m n vrq in the "B decay by which. n 

f fsio">ragn,ents\ecome stable nuclei, It ^JJWJJ.fjJ § + 1 
aeeav from a neutron- rich nucleus 2 to the ground state ot-4 + 1 
f'SrMdden. Then the nucleus Z+l is PJ^^^ft 11 ^ , 
U-+-* n - n fl n aY if It has sufficient energy, transiu^u v ' 

^f by^vSorating a" neutron. The neutron binding energy Is 
relatively small for these neutron- rich nuclei. It can, ol 

processes leading to the nucleus (Z+1J . aenDLeB e^v 

Triple Fission. Fission of a large nucleus Into three 
fragments--- two major fragments plus one alpha particle— Is 


FIG. VIII. 10 

Problem. Draw quantitatively as well as possible the curve 
FIG VIII .7, the energy of a nucleus as a function of some para- 
meter giving the extent of deformation. Invent a suitable 
nnwunp+PT to measure distortion. „^„„„ 

f It is probably simplest to describe the deformation by surface 

harmonics, I.e. , p _ a + 2> n S n 

1 + ■ „ vM + in^ raflltis and Sl, are surface harmonics; for 
where a is the initial radius ana ^ invarlance Is thus auto- 

SfcaliroroJldeg rorT Good Approximation to experiment Is 
obtained by Sling r = a + b 9 P 2 (cos6). The capillary energy is 

-ents are not joined, but are still close together, tney are 
deformed something like: QO The energy can be .rudely 
approximated by assuming each to be an ellipsoid The papers 
of Bohr and feeler, l,c . , Mel and Met nop oils 1 .c . «* , 
of Frenkel, J. of Phys. Aca d. Scl .USSR 1, So. 2 (1939) Un *n S _i^- J 
are pertinent to this problem. ^ 


rb 1 1 Model of the Nucleus 

H^gi 9 |SiSr^ nucleus In terms of nucleon orbits 
^J nv°» the ascription of the atom in terms of electron 
Sits, rnrorbi/olcture'ls valid If collisions are — -ough 
so that a nucleon may travel at least across the nuel •££**£* 
col lision. This requirement seems at first not to dc -u±±-l.l_ 
h;-As far as terms linear in the br 

J n 


In a v 

ucleus , 


is of 

the ord 

er o 


the saean f 

the nucleus . 



s , and 

th e s 


) When 


Nuclear Orbit Model 


at ?** 20 Kev the n-p scattering cross- sect ion 
f 0.3 barn, and, for the known density of nuc- 
ree path is only about l/j or so the radius of 
ever, there are two factors which this calculation 
e make the orbit picture appear not so untenable, 

ential well. If 
Spaced hue leons , 
blend together to 

2 ) The uucle 
energy states are 
occur between nuc 
ferring both the 
principle prevent 
state . Diagramma 

nuc 1 eon passes 

the nucleon is 

the wells may 

form a roughl 

us is a degene 
, for the most 
leons only if 
nucleons to em 
s two nucleons 
tlcally , 

another, It passes through a pot- 
constantly passing other closely 
be so closely spaced so as to 
j uniform potential. 

rate system in which the lowest 
part," filled, A collision can 
the collision results in trans- 
pty states. The Paul! exclusion 
of the same kind in the same 

Initial momenta 
of two nucleons 
plotted In mo 
entum space 

approximate be 
few, but some 

collision cause 

direction. Bot 
must represent 
states. If the 
states on the 1 
lision does not 
if occupied sta 
completely fill 
there could be 
mdary of filled 
unfilled states 

_, to change 

re 1 

ew end points 
prev 1 on sly empty 
re are no empty 
ocus A, the col- 
occur. Thus, 
tes described a 
ed perfect sphere, 
no collisions at 

state region; 

Neither of these tuo Ideas 
extent that they represent 
is justified. 

The orbit model has been explored 
absence of accurate information, a squ 
with rounded corners, is adooted as th 
The depth of the potential is assumed 
A. This Is justified by the computati 
showing that the kinetic energy of the 
^A*a , is, to fair approximation, inde 
the gas model. The requirement that t 
neutron be about 8 Mev then fixes the 

las been fully investigated . To the 
-.he true situation, the orbit model 

ith some success. In the 
are well, or a square well 
e form of the potential, 
not to change much with 
on at equation VIII. 48 

highest occupied state, 
pendent of A, according to 
he binding energy for a 
total depth of the well. 

Quantum mechanical calculations for the square well give 
levels which may be denoted as follov. r s : 

Is Ip Id 2s If 2p lg 2d 3s Ih 2f 3p 11 2g 3d 4s 

6 10 2 14- £ \1 id 2 2 2 (4 6: 2£> IS l° Z 





16 8 

The letter gives. X in the usual way, that Is, 
means ^ ~ ~5 , etc. The number gives the nu: 

r o. 



summed no . 
of occupants 

means^r = , f 
radial nodes. 

"l" means no node, "2" means one node, etc. (This differs from 

atomic spectra notation, 
n - J - 1. ) 

:ere the number of radial nodes is 

For a sharp cornered well , the states listed are about equal!] 
spaced In energy. When the corners Eire rounded off, the states 
shift so as to clump into the groups of states given in the list 

Ch. VIII Nuclear Orbit Model 169 

above*,* for example, Id shifts closer to 2s. 

As mentiored above, the depth of the well 1b approximately 
constant throughout the periodic table The radius R increases 
With A according to R = 1.5 X 10~H A /3 . For small radius, only 
the Is states are hound. As the radius Increases, hound states 
of higher It become possible, in the order listed above. 

The Paull exclusion principle prescribes the maximum number 
of one" kind of uarticle that. can occupy a particular space state. 
just as in electron configuration of atoms. There may be 2(2^+1) 
identical earticles in the state having angular momentum A . The 
factor 2 represents the two possible spin orientations. 

We may attempt to build up the species of nucleus out of 
nuclear shells, just as the periodic table is obtained in atomic 

P resumed configuration (P;K) 
il H 2 D (Is- Is) meaning a nroton in Is, a neutron in Is. Could 
' ' ' have spin or 1 and still accord with Pauli 

nrir.cinle. Actually observed spin 1. 

2) He 3 (Is 2 ; is) Spin 1/3 necessanily ; 

must be opposed. 

since the proton spins 

(lsjls"-) Spin l/2 necessarily, similar to above. 
(is 2 ; Is 2 ) Spin 0, a "closed shell" nucleus. 

3) lh> 
4.) He 

The is orbit for both neutrons and protons is now filled, 
The next orbit, lp, accomodates six neutrons and six protons, 
Giving 1 " as the next closed shell nncleus. 

5) He 5 

does not exist; evidently in He 5 , (is ;ls"lp), the 
It? state is not bound." Hq5 is the compound nucleus 

* This may be understood by the following argument . Consiaer a 
sharp cornered square well having ei gen state functions Y ± : lhe 
perturbation In energy of state k when the corners of the well are 
rounded can be found "from first order perturbation theory, ««■«£ 

J- as the unperturbed states, 
shift in energy is 


■potent to. 1 . 

= cJj4^M^> 

.c = fir, 




where ix* is the radial part oif-vf* , 
H Is the perturbation of the poten 
tial that occurs when the corners 
are rounded . Let H = H-, + &* . H 


gives the rounding on the inside, at. 
A, H 9 gives the rounding outside, 
i p r m at_B. „ j& 

H 2 Is negative, but c| Kl^ > C L \AA*& ] therefore AE is +. 
Now the reason that states cf higher angular momentum have larger 
AE's Is that particles in these states spend relatively more 
time near the edge of the well, in region A. 


Nuclear Orbit Model 


for the reaction n + He 4 — *-jj + He 4 . A scattering 
resonance Is observed at about 1 Mev , presumably 
representing the lp level of KrP . In fact, it 'is 

The sb 

claimed that two peaks are observed. 
p l/2 and %/g 

would be the p., /n and p^ ^ states; 

an He 

(Hall and Koonta, Ph 

. Rev , 7k 

6) ti l 

i 2 2 
(la ,lp-;ls ,lp) 


of n 

; 196) 

The end 

this "period" is 

There are two independent systems., 
each consisting of a soln 1 (orbital) 
and of a spin l/2 (intrinsic spin). 
There are many possible total angu- 
lar momenta, including the observed, 
namely l . 


the next closed shell 

31! 3. 

It has spin and is unusually stable 

On the orbit model as given so far, the closed spoils for 
either the proton or neutron confl durations are at 2, 8, 20, 40, 
70, 112, 166,.... neutrons and/or protons. For l&vr.e nuclei the 
Coulomb energy makes the number of proteins less than' the number 
of neutrons, and the nucleus cannot have closed shells of both at 
the same time . 

VIII. 64 
of neutrons, 

Empirically, the closing numbers appear to 
2, 8, 20, 50, 82, 126 (Magic numbers) 

Nuclei having either such a number of protons, c 
are unusually stable. 

These are not the same as the closed shell numbers given bv 
the preceding development. This discrepancy has been Interpreted 
by M. a. Mayer as follows. Suppose that spin-orbit coupling splits 
the energy levels corresponding to different J values, that is, 
lg splits into lSoy 2 and l®^'. Assume that the level with larger 
J is mono stable, i.e. lies lower. This assumption is not contrary 
to any known facts about the nucleus. Then the former closed shell 
number 40, for example, must be alt ©red as follows: 

Old dividing line 
between shells 

(Higher i numbers are 
a a erased to have larger 

5~0 ]nuc|eans 
u|» to here 

Hew dividing line between shells 


d. Mayer, Phys.Rev . 7_4 235 (1948) 

Ch. VIII Radiative Capture H(n,"tf )D 

The next shells are 

1&7/2 2^ 3s lh, 


T.l/2 1 

82 nucleons to here 
lh g/2 2£ 3P lt 13 / 2 , 

126 nucleons to here 

Thus the altered nuclear shell theory Gives closed shell numbers 
that anree with the numbers deduced from experiment. 

Problem. Look up the nuclei having; closed shells of eitner 
neutrons on protons, and note to what extent they tend to toe 
relatively stable . 

L. Capture of Slov.- Keutrons by Hydro gen 

This nu clean reaction is one of the few that can be calcu- 
lated with some precision. 

Only the S wave comoonent of the resolution of the neutron 
plane wave into radial functions is important for the low neutron 
enerrj.es to be considered here, i.e., thermal energies, ^ > O 
waves have angular m omen turn with respect to the proton so great 
as to keen a slov: neutron beyond the reach of nu clean forces. 
(See chanter VI, section c, , p.ue-rtj The only source of angu- 
lar momentum is intrinsic spin. The final state Is the deuteron 

3^ state (the 1 S state Is not bound). 

n(slow) + H ; — >• D + a ( itiw = 
The conceivable transitions are 


.19 Mev, 

"S (continuum) 
■^S (continuum) 

( magnetic dipo'Ls)^ 3g 
■ ->- 3 S 

(is not possible, 
a s shown just 
after eq. VI 1 1. 78) 

The parity selection rale excludes electric dipole for S-> S 
transitions, Ch. V, p. loo . 

The physical mechanism, 
in classical terms, is the 
following. Since the magnetic 
moments of the proton and neut- 
ron are not equal , ^p ~P J^h j 
the totaly^f for the deuteron 
does not have the sane direc- 
tion as the annular momentum 
vector, FIG. VI II. 11. There- 
fore the .system is a rotating 
magnetic dipole. The formula 
for magnetic dipole radiation 
is almost identical to that for 
electric dipole radiation, 
differing only In that the 
magnetic "moment matrix element 
Is used (see Ch . V, p. 95 ) 



Moment* j 

^ Y-W 

resultant magnetic 
moment moves around 
the fixed I axis 

FIG-. VIII. 11 



Radiative Capture H(n, t )D 

Probability of transition ^ ^ \ u, \ t,,,~r ,-- 

7 unit time =^X^S l/NWl.Jwfad I VIII. 65 


where W = (energy of transition)/^.. The energy of the transition 
is 2.2.3 Mev, the binding energy of the deuteron, provided the 
energy of the incident neutron is sraall . 

The wave functions .are 

[final) T(%) = 

., j( (+-) + (-+) )l/j? s z 

s a = + i 

(initial) % } s ^V( + _)^-+3]% yiIIt66 

Notation for the spin functions: (+-) siear.s proton spin up,(+)j 
neutron spin down, (- ) . The space parts of the wave functions 
are independent of angle, being S functions. 

u(r)/r is the solution of the Schrodinger equation for the 
known potential, U-* , the triplet S state neutron-proton potential 

The radial equation is 

M-U) + T^(^)(\W 3 -U 3 !^)) J W = 

VIII. 67 

W-, = -2.23 Mev; U4 - -21 Mev for r smaller than the range of nuc- 
lear forces, and beyond. (See theory of the deuteron in Ch.vi , 
p. us .) u(r) has the form: 

u(r) c£ © ' for r>R. k* - \ YmV#J - 2.26 x 10 12 cm" 1 

** -hit 

The error committed in using u(r) = G g 3 for all r is tolerable, 
since R < l/k v and we shall correct partly; as follows. When 
u(r) is set — C, 6" 5 , we get somewhat too large values in integ- 
rations over r. If we compute the normalization C using the same 
approximation, an error is made which tends to compensate. Normal- 
izing so that , f * ..j 

fir = 4TC Me 5 | d& = 1 

we get 

VIII .68 

VI II. 69 
VIII .70 

j(r) is the S wave component of the resolution of a plane 

wave Q-~* s into polar eigenf unctions, but perturbed by the poten- 
tial well of the proton, The potential well is U-, , the potential 
existing between a neutron and a proton in a sir. "let S state. 

Q* & is already normalized to one particle per u""it volume in 
the absence of the 'oertnrbinr. ootential well, The unperturbed 

plane wave can be exuanded as follows* 

&K IM^)/&)"LN Pit— ) 

VIII .71 

*3tratton, "Electromagnetic Theory," p. 408 


The S wave Is 

Radiative Capture H(n, T )D 


-) 4N 



VIII. 72 

We must now find how U perturbs this. The m mature of the 
wave function is areat+y incr eased in the region of the well 
and is nearly Independent of neutron enor-y, "since V-c neutron 
energy is small compared to the well de^th". From i'ib theory of 
scattering of neutrons by nrotons, Ch . VI, m Itnow t'^t for si ok 
neutrons the tangent to the radial function ) (r) at the noter,- 
tial well ed-e V, intersects the axis at a distance a-, from r - 
Therefore ;.{r) must have the form: 

W =j(x) 


tje^mntn^ o"f tne function — - 

in fe 

It Is evident franj the fi-are that j ( r )« (l/h) sin(kr + ia.,0 
a fairly good approximation to the tru 
for small r. 

Computation of the matrix element 

t * 
! fimt 

j (r) . j {r}^ JajJ + r 

The matrix element is 

VIII. 74- 

The marnetic moment vector is 

VI I I. 7? 

where C T etc . are the Paull spin operators for proton and neutror 
These operators act as follows: 

<X x (+)^(~) 

a(+)^i(-) cr„(+) — (+) 

i 6 


For example, C^ ( + _ ) _ („) f le£vlnr> thG lzeutTOn sr , ir , B ymtol u^- 
cna n red . We c al c a 1 a t e (jj^) a s a n ex ampl e . 

For this we need J^<|^ +ja n CT 71 ) , the spin parts of which -ive 

r nw — / 


3 ^ ^ / \7. 


(Note t hat if Jj F - ^J N the matrix el em ent would be 0. H>- sleally 
this corresponds to the case that the resultant ma -net! c moment 
has tne same direction as the an^lar momentum, Which is invariable 

1V4 ' — -— Ba&latlye Capture H(n, 5 )D Ch. VIII 

and the average magnetic moment would be constant,) 

The term above has the same form as the spin function for the 
final 3 Z - state, and is multiplied by 



i> = i^ [c+-)+£-fn^ s,-o 

Due to orthogonality of the spin functions, only the ¥ 
S z = gives a contribution. Therefore 




VIII. 78 

It can be seen now that -^S (continuum) 

''S (bour.d) does 

not occur, for in such a transition the matrix element depends 
on the integral over two S wave functions for the same potential, 
and this is 0, by orthogonality of eigenfunctions of the same 
Hamiltonian. In the transition considered above, 1 S — 1-^3, the 
3 functions are for different potential functions. 

Since there is no preferred axis, \(df.J)^\ 2 =^f./h\ l ~\^li)iJ\ 

\M U V=3Wi,)S TOI - 79 

from VIII. 65, 

.From the definition of the cross section & , a 
we get 


i#r-rAd V ji*}Mfifa 

VIII .80 

where a factor l/4 comes from the fact that only l/4 of the ini- 
tial continuum states of the neutron and proton are singlet S. 


Using the values: I13 = 2.26 x 10 



a l 

CA K 10 

2.32 x 10" 12 





VI II. 81 

VIII .82 

§W velocity (f%j 

For thermal neutrons, V = 2 ..2 x 10 5 cm/ sec, 0^*} = 0.29 barn. 

The experimental value is O.30 barn. An elaborate theory includ- 
ing tensor forces gives 0,31 barn. 

Remark s on neutr on canture in 1 i ght nuclei. C^caVM is in the 

range .001- .01 barn, with some exceptions, 

H, with c^ 


= »% 



Photonuclear Reactions 


jvL°£2 f + them - p is rela tivei y meb a ls due to fche fact that 

:!.:^ dCU !'f ro r- State is alcost bo1 -^, i.e., because I aj is 

exceptionally large 
light elements, but 

. There are other cases of large.-*** in the 
toy the reaction ( n , pa rt i c i e ) : 
B^O C( n ,a) - 3800 barns 
He?, 0"(ii,p) = 6000 


f.3 cr( a , a 

-*. , - !«.»«, i - 8000 
W 14 , C(n,p) = 1.7 

+ ?\f 0r fI ' J? SfS'll, compared to the others, because, in contest 

factor ISf^f i ? 8M * 8 ' + V 8 larG6 en ° U ^ t0 Eake *S barrief 
iactor ejcpt-g-j » important for emission of a charged uarti-cle 

IftEfSSer'or g*.**"^ ° f * ^on is reduced^ fgcSr" 

M. Fhotonuclear Reac tions 

The Be are :nainiy"rY7n), althou^ U.2r_), (J.fj, etc are 
hnown also. The threshold for (jr.n? is the binding energy If 

«°° n ^ USUal ? ab ° Ut , 8 Mev " ^eptions te this figra are 
■■■"■-■■ : thresholds of 2.2 and I.71 Mev, respectively 

Strong J absorption resonances have been observed 

at 30 Mev ( t h re sh o 1 d 18 . 7 Mev } , 



2^^l; at 22 Key, and in „^Ta 
at 16 Mev, Goldhaber and Teller *» have interpreted these res- 
onances as due to an electric dipole interaction in ,M '\b the 

neutrons." Btt ° 1 * lB mm a ' S a Unit with ro ^ ect ^ aS the 

Phot gd i s 1 nt e ,-rs.t io n of the de utero r 
pliotonuclear reaction having a reasonably 
The transition if + D — *-n + p (unbound.) 

This is the only 
reasonably detailed theory. 

Except at 



J S 
3 Q 

(33 .ground 
(D ground 

p (unbound) occurs in two ways: 

3G^r.ej -*- b I continuum) 

sl c»n e ) -^ s- ^p [ ooati nuum ) 


tenths of an Mev above 

to be 

- - ---, —.Av^tjiwo-g \Li_.j to a fe 1 .-: tenuns or an 

threshold), the latter is the dominant reaction. The 
momentum of the P state requires too high a velocity 
attained by lov,- energy neutrons. 

rp , p + ^ fjr* m °f' 3 S— ^(continuum), is the inverse to the 
reaction discussed in the ore ceding section. Therefor© its 
cross section is obtainable from detailed balance arguments, 

allied e #*£ be va^VnTJ^ 6aUatl ° n VIII - Bl ls ^ st *&»*' 
iUocl t0 bs valid ^or ail neutron energies. The result is 

S .U - Vi* ^ ^ = H' EK + W ' 

f,(j +W.) 


where Cu D is tbe angular frecuercv at threshold, namrdv 
finding energy W, and Wo is the magnitude of the fictitious 
binding, energy of the deuteron sin -let state **** " " " 

* For definition of & see VII 1, 12 or I II. 3, o 58 
^ Goldhaber and Teller, Phvs.Rcv. 74 1046 fl948). 
**"* Bethe , D , p . 58 " " 

***•* W Q is really a parameter such that 
state were bound, its energy would have to be W 
tne observed singlet scattering cross section, 
FIGURE in this footnote, which is continued on 

if the singlet deuteron 

in order to 
Shown on ti- 
the next page 




Photouuclear Reactions 


The cross section for the second tvne of transition 3s-»-3p 
can he computed usiri" the formula for G amma ray c^sorptiiii.:* 

CT^^^lMp VIII. 84 

^a la the reduced mass, ano v the velocitv of the emitted 
particle. M is the matrix element for the electric dipale moment 

M -sr«* i »'^8* 

VIII .85 

£»« ^ coordinate of the proton with resect to the cento- of 

ST+H^ 6 + faCt ? r tf? rssulta because the neutron is uncharged, 
ano tiie system is lihe half a dipole . 

Wis will not carry out the calculation, taut merely describe 
the approximate wave functions. 

Except when hi rh est accuracy is wanted, T, 3 „. may he featoan 

(fro m e qua 1 1 o u VI 1 1 . 7 ) . In t-h e absence of the umlear got ©Btlal 
*(3p) is ths P wave component of a plane wave (see equation 
VI II. 71), and has the form I {$*** ^i^ ^^ + , „ 

.SaITCl &&-#*-} iron tne properties 

^ x ^ a -^-°Et d Vessel functions.** At loir energies, (A) ir FIG. 
vi_i.it;, Tp p j is nearly in the reel on of the short range 

nuclear potential well-, and Is therefore little disturbed by 
it, una is like tne P wave of a free particle. At hlrdier ener- 
gies ¥ ( 3 p ) lias shorter period, and is Perturbed m 0re by the wall 
(B) la FIG. VIII.12. 



r¥ is plotted in FIG. VIII.12 because the behavior of I is seen 
best when matrices are written in the form, ^cc Unf*) f (vif;) ch. 
The result is: ^ 

U CS» f 7 3 M cOj d J Uj 2 - j VIII .86 

where UJ 6 is the frequency at threshold, as before. The two cross 
sections e^Oiwfctffc) and ^^(etetii^are plotted In FIG. VIII. 13 


^ " 


* Pethe D, ■ 
** Schiff, n". 

— a, 

. 56 

fictitious Uond state T^ _^ 
Wing same Qi j nud eneig/ Wo. 

Very High Energy Reactions 



of If 

The reason that Cfy^fw^) rises so sliarply is that energy above 

the threshold (0 excitation) is almost a resonance energy. Ac 
can be seen from equation VIII .83,, if W q were zero, there would 
be a resonance at excitation energy. 

g. Remarks or Very Hirfi Enerr.Y Phenomena 

- a) Stars . A rroninent feature of very high energy phenom- 
ena is the production of stars. These are seen in photographic 
emulsions and in cloud chambers. The prongs are due to ionizing 
^articles, protons, albhas, or larger fragments. Neutrons leave 
no trace. The quantitative interpretation is rudimentary. Star 
production car be interpreted roughly as evaporation of particles 
from a very high temperature nucleus. M. Goldberger*- has added 
to the evaporation model consideration of the situation when a 
very high energy nucleon, of energy around 100 Mev, has first 
entered the nucleus. At this tine, when its wavelengh is very 
email and its energy is still undistributed 
among the other nucleons, it has collisions 
which knock other nucleons out of the nuc- 
leus immediately. Soon, however, the energy 
not curried off by these quickly escaping 
nucleons is distributed in the form of 
statistical excitation of the nucleus. 
Then the escape of nucleons is described 
by the slower arocess of evaporation. Goldberger calculates that 
about 1/2 of the initial 100 Mev leaves immediately t and about 
1/2 is trapped and produces evaporation. 

b) Deuteron st r 1 y, ping ■ ** Deuterons, given an energy of 200 
Mev In the Berhelev 184" cyclotron, impinge on a target Inside 
the cyclotron, and" a beam of very high energy neutrons emerges. 
(FI&. VII I. 14 J The interpretation is that the proton of the 
loosely bound deuteron hits and is caught by a nucleus, and the 
neutron flies on. Ideally, the neutron would have about 1/2 the 
energy of the deuteron. However the neutron has a velocity with 
resnect to the center of mass of the deuteron. After the proton 
is removed, the neutron's total velocity vector is the vector sum 
of the velocity of the center of ma s s of the d eu t e r on and th e v el - 
ocitv of the neutron relative to the center of mass at the Instant 
the proton is removed. Therefore the neutrons emerge with a 
spread in energy and in angle . 


* M.L. Goldberger, Phys.Rev . Jh 1269 (1948) 

** Helrr.olz, McMillan and Sewell, Phy s . Rev . jS 1003 (1947) 


Very High Energy Phenomena 



tjeuteron ^am 

CYe.UtVW (of at on 

FIG. VIII. 14 

FIG. VI II. 15 

e;) b s e rv ocl exchange 1 ri p- n 3 c a 1 1 e r i n g . Wh e r. a be am of 
350 Mey protona hits a scatterer placed, inside the Berkeley cyclo- 
tron, FIG. VI II. 15 , high energy neutrons (as well as protons) 
emerge. This shov.'s that in many collisions the neutron and 
proton exchange roles: 3ccttie rer 

^ ^__ . ^- R^jftyon 

A similar effect has been observed in ft-.-rj scattering.'* 

Problem: Compute the distribution in energy and angle of scat- 
tering of neutrons re suiting from the stripping of fast deuter- 
ons , using the simple mod~l described above. 

(This calculation Is done in the first part of Berber 1 s paper 
on the theory of stripping of deuterons, Phy a ■ Rev . 7_2 1007 
(l94y). Although the most probable neutron energy is l/2 that 
of the deuteron, the energy spread, is larger than the binding 
energy of the deuteron by a factor of (l/2 E d /EE(d) ) /i «{ 100/2 y z 
ar 7. This may seem like violation of cor.sen.rat ion of energy, 
Houever, the velocity of the neutron relative to the laboratory 
system = '^^_ elx ^ t + ^ rel _ c i eu t. > and - t - le elle - T CJ °f the neutron 
In the lab. system thereforeis 

^M(aij 1 + fe.^ifed^O^ + Zarj^^.fad-t) 

:ere the last term, is the marl 

contribution to the snread, 

For a su mm a ry of very hi gh e ne r f-y phenomena, s e i 
Chew and Moyer, Are. J. Physics IS 1S5-1J5, Ig 17, 22 20 5- 

*This is discussed further under "Exchange Forces" in Ch. VI, p. 122